Why Markov chains, Brownian motion and the Metropolis algorithm
- We want to study a physical system which evolves towards equilibrium, from given initial conditions.
- We start with a PDF \( w(x_0,t_0) \) and we want to understand how the system evolves with time.
- We want to reach a situation where after a given number of time steps we obtain a steady state. This means that the system reaches its most likely state (equilibrium situation)
- Our PDF is normally a multidimensional object whose normalization constant is impossible to find.
- Analytical calculations from \( w(x,t) \) are not possible.
- To sample directly from from \( w(x,t) \) is not possible/difficult.
- The transition probability \( W \) is also not known.
- How can we establish that we have reached a steady state? Sounds impossible!
Use Markov chain Monte Carlo